Graphing Parametric Equations: Learn It 2

Lumen Learning, OpenStax

Graphing Parametric Equations: Learn It 2

Rectangular vs. Parametric Graphs

Graph the parametric equations [latex]x=5\cos t[/latex] and [latex]y=2\sin t[/latex]. First, construct the graph using data points generated from the parametric form. Then graph the rectangular form of the equation. Compare the two graphs.

Construct a table of values like the table below.

[latex]t[/latex]	[latex]x=5\cos t[/latex]	[latex]y=2\sin t[/latex]
[latex]\text{0}[/latex]	[latex]x=5\cos \left(0\right)=5[/latex]	[latex]y=2\sin \left(0\right)=0[/latex]
[latex]\text{1}[/latex]	[latex]x=5\cos \left(1\right)\approx 2.7[/latex]	[latex]y=2\sin \left(1\right)\approx 1.7[/latex]
[latex]\text{2}[/latex]	[latex]x=5\cos \left(2\right)\approx -2.1[/latex]	[latex]y=2\sin \left(2\right)\approx 1.8[/latex]
[latex]\text{3}[/latex]	[latex]x=5\cos \left(3\right)\approx -4.95[/latex]	[latex]y=2\sin \left(3\right)\approx 0.28[/latex]
[latex]\text{4}[/latex]	[latex]x=5\cos \left(4\right)\approx -3.3[/latex]	[latex]y=2\sin \left(4\right)\approx -1.5[/latex]
[latex]\text{5}[/latex]	[latex]x=5\cos \left(5\right)\approx 1.4[/latex]	[latex]y=2\sin \left(5\right)\approx -1.9[/latex]
[latex]-1[/latex]	[latex]x=5\cos \left(-1\right)\approx 2.7[/latex]	[latex]y=2\sin \left(-1\right)\approx -1.7[/latex]
[latex]-2[/latex]	[latex]x=5\cos \left(-2\right)\approx -2.1[/latex]	[latex]y=2\sin \left(-2\right)\approx -1.8[/latex]
[latex]-3[/latex]	[latex]x=5\cos \left(-3\right)\approx -4.95[/latex]	[latex]y=2\sin \left(-3\right)\approx -0.28[/latex]
[latex]-4[/latex]	[latex]x=5\cos \left(-4\right)\approx -3.3[/latex]	[latex]y=2\sin \left(-4\right)\approx 1.5[/latex]
[latex]-5[/latex]	[latex]x=5\cos \left(-5\right)\approx 1.4[/latex]	[latex]y=2\sin \left(-5\right)\approx 1.9[/latex]

Plot the [latex]\left(x,y\right)[/latex] values from the table. See Figure 4.

Graph of the given ellipse in parametric and rectangular coordinates - it is the same thing in both images.

Next, translate the parametric equations to rectangular form. To do this, we solve for [latex]t[/latex] in either [latex]x\left(t\right)[/latex] or [latex]y\left(t\right)[/latex], and then substitute the expression for [latex]t[/latex] in the other equation. The result will be a function [latex]y\left(x\right)[/latex] if solving for [latex]t[/latex] as a function of [latex]x[/latex], or [latex]x\left(y\right)[/latex] if solving for [latex]t[/latex] as a function of [latex]y[/latex].

[latex]\begin{align}&x=5\cos t \\ &\frac{x}{5}=\cos t&& \text{Solve for }\cos t. \\ &y=2\sin t&& \text{Solve for }\sin t. \\ &\frac{y}{2}=\sin t \end{align}[/latex]

Then, use the Pythagorean Theorem.

[latex]\begin{gathered} {\cos }^{2}t+{\sin }^{2}t=1\\ {\left(\frac{x}{5}\right)}^{2}+{\left(\frac{y}{2}\right)}^{2}=1\\ \frac{{x}^{2}}{25}+\frac{{y}^{2}}{4}=1\end{gathered}[/latex]

Analysis of the Solution

In Figure 5, the data from the parametric equations and the rectangular equation are plotted together. The parametric equations are plotted in blue; the graph for the rectangular equation is drawn on top of the parametric in a dashed style colored red. Clearly, both forms produce the same graph.

Overlayed graph of the two versions of the ellipse, showing that they are the same whether they are given in parametric or rectangular coordinates.

Graph the parametric equations [latex]x=t+1[/latex] and [latex]y=\sqrt{t},t\ge 0[/latex], and the rectangular equivalent [latex]y=\sqrt{x - 1}[/latex] on the same coordinate system.

Sketch the graph of the parametric equations [latex]x=2\cos \theta \text{ and }y=4\sin \theta[/latex], along with the rectangular equation on the same grid.